Nuclcar Physics B274 (1986) 429-447 North-Holland, Amsterdam

THE UNITARY S U P E R M U L T I P L E T S OF d = 3 ANTI-DE SITTER AND d--- 2 C O N F O R M A L SUPERALGEBRAS*

M. GfSNAYDIN

Unit'ersi O" of California, Lawrence Lwermore Laboratoo', Livermore, California, 94550 and Lawrence Berkeley Laboratory**, Berkeley. California. 94720, USA

G. SIERRA

Facultad de Ciencias Fisicas. Unioersidad Complutense. Madrid, Spain

P.K. TOWNSEND

DA MTP, Unit,ersi O' of Cambridge, Cambridge. England

Received 27 January 1986

Wc classify the d ~= 3 anti-de Sitter and d= 2 conformal superalgebras and give a general construction of their positive energy unitary representations. The corresponding supergroups are in general direct products G~ x G_, of two simple supergroups G~ and G 2. We study in detail the cases when G l (and/or G2) is SU(1,1/N) or Osp(2M/2,R).

I. Introduction

G r o u n d s ta tes of supergravi ty theories that preserve supe r symmet ry have a

s u p e r g r o u p as their mani fes t invar iance group, i.e. par t ic le states be long to un i t a ry

s u p e r m u l t i p l e t s of this supergroup . The superg roups of re levance in this context

have been c lass i f ied by N a h m [1]. The an t i -de Sit ter supergroups , in par t icu la r , have

a t t r a c t e d cons ide rab l e interest in recent years. These are supe r symmet r i c genera l iza-

t ions of the an t i -de Si t ter groups S O ( d - 1,2) of ( d - 1) + 1 = d space- t ime d imen-

s ions. Simple an t i -de Si t ter (ADS) superg roups exist only for d = 2, 4, 5, 6 and 7.

O b s e r v e that d = 3 is not r epresen ted in this list, desp i te the fact that there does

exist a d = 3 supergrav i ty theory with a supe r symmet r i c (ADS3) b a c k g r o u n d [2]. The

reason is tha t the A d S 3 group SO(2,2) is not simple, but is in fact local ly the

* This work was supported in part by the US Department of Energy under contract W-7405-ENG-48 and the Director, Office of Energy Research, Division of High Energy Physics of the US Department of Energy under contract DE-AC03-76-SFO-0098.

** Participating guest.

0550-3213/86/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

430 M. Giinqpdin et al. / d = 3 A d S and d = 2 {'onformal superalgebras

product of two SO(2,1) groups.

S0(2, 2) -= S0(2, 1) x S0(2, 1).

The group SO(2, 1) is isomorphic to the following groups

SO(2,1) ~ SU(1,1) ~- SL(2,n) --- Sp(2 ,n ) .

(1.1)

The groups S O ( d - 1 , 2 ) are also the conformal symmetry groups in ( d - 1 ) - dimensional space-times. In any conformally invariant supergravity theory the spectrum of states falls into unitary representations of the conformal group or a super-conformal group if the corresponding ground state is supersymmetric. In 2 dimensions the conformal group is infinite dimensional and its finite-dimensional subgroup is simply SO(2, 2). (Below the term-two-dimensional conformal group will always refer to the finite-dimensional SO(2,2).)

In this article we shall study the "positive energy" unitary representations of the d = 3 anti-de Sitter supergroups or equivalently the d-- 2 conformal supergroups. In addition to its importance for anti-de Sitter supergravity theories in d = 3 and conformal supergravity theories in d = 2 the knowledge of the unitary supermulti- plets of the d = 2 conformal superalgebras is important for superstring theories. The background field theories that describe superstring propagation are required to have conformal symmetry [3]. We hope that the unitary supermultiplets given in this paper will lead to formulation of new background superstring theories.

Since the group SO(2, 2) is not simple, the corresponding space-time supergroups CJ (d = 3 AdS = AdS 3 or d = 2 conformal = Con2) are in general direct products of two simple supergroups G and G'

{ ~ = G x G ' .

The supergroups G and G' can be any of the following

(i) O s p ( U / Z , R ) D O ( U ) X Sp(2, R),

(ii) SU(N/1,1) DU(N)XSU(1,1), N~2

o r

su(2/1,1)

(iii) Osp(4* /2N)

(iv) G(3)

(v) F{4)

(vi) D1(2, 1, a)

D SU(2) X SU(1,1),

D 0*(4) x Usp(2N) -- SU(2) × Usp(2N) X SU(1,1),

D G 2 x SU(1,1) with G 2 compact,

D Spin{7) × SU(1,1) with Spin(7) compact,

D SU(2) x SU(2) X SU(1,1). .3)

(1.2)

M. Gffnm'dm et al. / d = 3 A d S and d = 2 collfi~rmal superalgehras 431

The supersymmetry generators of AdS 3 (or Con2) supergroup above transforms as the (1/2,0) ~ (0,1/2) representation of SO(2,2) - SU(1,1) × SU(1,1). On the other hand there exist simple supergroups whose non-compact even subgroup is simply SO(2, 2) namely

SU(1,1/1,1) D SU(1,1) × SU(1,1) ,

D2(2,1; a) D SU(1,1) x SU(1,1) × SU(2). (1.4)

However in these supergroups the "supersymmetry generators" transform like the vector representation (1 /2 ,1 /2) of SO(2, 2). The supersymmetric theories based on these supergroups would have to satisfy some unusual spin and statistics connec- tion.

The positive energy unitary irreducible representations (UIR) of the (ADS)3 supergroups fall into three classes:

(i) singleton (or doubleton) irreducible representations (irreps), (ii) massless irreps, and (iii) massive irreps. We shall construct these irreps using the oscillator method [4,5]. The massless

irreps are of use in the construction of d = 3 AdS supergravity theories and the massive irreps would be relevant in the context of a Kaluza-Klein compactification of a higher dimensional supergravity to AdS 3 times some internal space. Such compactifications of d = 7 supergravity have been found recently [6]. The modes on the compactifying space fall into massless and massive irreps of the relevant AdS 3 supergroup.

The singletons (or doubletons) generally decouple from the spectrum in such compactifications, being pure gauge modes, and this is not surprising as singleton (or doubleton) irreps have no Poincar6 limit. However considered as supermultiplets of d = 2 conformal supergroups these singleton irreps play a special role. The (ADS)3 supergroup acts as the superconformal group on the boundary S ~ x S 1 of the d = 3 AdS space. One can construct field theories based on the singleton irreps that live on the boundary of the AdS space. Such theories can be interpreted as conformally invariant field theories in one lower dimensional space-time [7]. The singleton field theory for (ADS)3 is therefore a d = 2 conformally invariant field theory. A study of the relation of singleton field theories to strings and superstrings will be presented elsewhere [11].

2. The positive energy unitary representations of SO(2, 2)

The group SO(2,2) has a linear action on a four-dimensional space M 4 with signature ( + - - + ). We shall denote the coordinates of M 4 as x °, x z, x 5, x 6 such

432 M. Giinqvdin et aL / d = 3 A d S and d = 2 conformal superalgebras

that its metric 3~M N is diagonal and

7~00 = '066 = 1 ,

nn =rt55 = - 1 . (2.1)

The generators JMN of SO(2, 2) satisfy the commutation relations

[ JKc, JMN] = i( ~KNJLM + ~LMJKN- ~IKMJLN- ~LUJKM),

K , L , M , N = O , 1 , 5 , 6 . (2.2)

The generators JKr Of SO(2, 2) can be decomposed into generators of two commut- ing SO(2, 1 ) subgroups [8]. We shall choose the following linear combinations of JKL as the generators of two mutually commuting SU(1, 1) subgroups:

I L?= :( J+, + Jol),

L~ = ~(J05 + J ,6) ,

1 L," = +(J+ , - J0,),

L R ----- ½ ( J 0 5 - J 1 6 ) ,

L L ' (J¢,6 + J15), LR= L = 5 ( ' / 0 6 -- g 1 5 ) . 3 5

They satisfy the commutation relations

[LL.L L] = - i L L , [L~,L~] = - i L ~ ,

[LL, L L] =iL L , [ L~, L~] = iL~,

[ LL, LLt ] =iL L ,

(2.3)

Considered as a generator the AdS 3 group the operator J06 goes, in the Poincar6 limit, to the generator of time translations. Similarly, the operator J15 generates the rotations. Hence we shall refer to the eigenvalues of J06 and J15 as energy E and spin S. Since Jo6 ~ L = ~(L 3 + LR), the positive energy unitary representations of SO(2,2) will be given by those representations of SU(1, 1) for which the generator L 3 has a spectrum bounded from below. These are precisely the lowest weight representations belonging to the analytic discrete series [5]. The unitary representa- tions of SU(1,1) are well-known [9]. The unitary representations of SO(2,2) have also been applied to 2-dimensional conformally invariant field theories [8]. Our aim in this section is to summarize the construction of the lowest weight unitary irreducible representations of SU(1, 1) by the oscillator method.

[LLi, L R] = 0 , i, j = 1 ,2 ,3 . (2.4 3

M. Giinaydin et aL / d = 3 A d S and d = 2 co~lformal superalgebras

The Lie algebra of SU(1,1) in a split basis takes the form:

433

[L °, L + ] = L + ,

[ L °, L - ] = - L - ,

[L +, L-] = - 2 L °, (2.5)

where L ° = L3, L + = L t + iL2, L - = L 1 - i L 2. (The L i stand for L~ or L~.) There are two singleton irreps of SU(1,1) and they can be constructed over the Fock space of a single oscillator. This is achieved by representing the generators of SU(1,1) as bilinears of a single boson annihilation and creation operators c, c*

_ = ' L ° ¼ ( ~ * c + ¢c*), L + = ~ c t c *, L - ~cc , =

[c, c t] = 1. (2.6)

There are only two states in the Fock space which are annihilated by L - and have definite L ° quantum numbers, namely the vacuum [0) and the one-particle state ctl0>. Taking these as the lowest states and acting on them repeatedly by L * operators one generates an infinite set of states which form the bases of the two singleton irreps:

I0), L ÷ I0) = ~(¢*)210), L + L + I0) . . . . .

c*lO), L+c*lO) : ½(c*)310), L+L+c*IO) . . . . . (2.7)

Note that these irreps are uniquely determined by the lowest states and hence they can be labelled by their L ° eigenvalues which we denote as l 0

L°l O) = ¼10) ~ l o-- ~,'

_ 3 L°c*l O) = 3c*10) ~ l o - ~. (2.8)

The singleton irreps of SO(2, 2) is then obtained by tensoring the singletons of two SU(1,1) factors. They will be labelled by the l0 L and l0 R of the lowest states of the singletons of these SU(1,1) factors.

IO) ® IO) = (tk, to ~ ) = ( ~, ~ ) ,

cl io) ® lo) = (~k, ~g) = (~,-~),

to) ® c*~10) ~ (~0 ~, 6, ~ ) = (~, ~),

cL lO)® cklO) ~ (Z,L,/,~) = ( I , ~),

(Eo, So) = (~,o),

(Eo, So) = ( i , ~),

(Eo, so) = ( i , - ~ ) ,

(Eo, So) = (~,o), (2.9)

434 M. (;fimn'din et al. / d = 3 ,4dS and d = 2 conJbrmal superalgebras

where the AdS~ energy and spin (= helicity) of the lowest states are labelled as E o and S..

Based on the experience with the higher-dimensional AdS groups we shall define the massless representations of SU(1,1) as being those that can be constructed using a pair of bosonic oscillators. In this case the generators of SU(1,1) are bilinears of tim form:

where

L + = a~b ~, L - = ab, L ° = { (a ta + bb t ) , (2.10)

[a, a t ] = [b, bt] = 1.

In the tensor product of the Fock spaces of the two bosonic operators there exist infinitely many states that have definite L ° quantum numbers and are annihilated by the L - operators. They are in general of the form:

o r

(a*)"10) - In,0>, n = 0 , 1 . . . . .

(b*)"'10) _= 10, rn), m = 0 , 1 . . . . . (2.11)

Choosing any of these states as the lowest state 1[2) one can generate the basis of a unitary irreducible representation of SU(1, 1) by repeated applications of L + oper- ator. For example, if we choose ] 9 ) = (a'~)"10)= In,0) then the infinite set of states

Isa) = (a t )"10) = In ,0 ) ,

L+ Ira> = I n + 1,1>,

(L+)21~2) = In + 2 ,2) ,

(2.12)

form the basis of a UIR of SU(1, 1) labelled by the L ° quantum number of the lowest state which is ~(n + 1).

The "massless" positive energy UIR's of SO(2,2) can be obtained by tensoring the massless irreps of SU(1,1) x- with those of SU(1,1) R. They are labelled by the (/~, l~), or (E o, So) of the lowest states. We give below the possible lowest states

and their labels:

In,0)k ® Ira,0) F,,

( l t o , l oR)=(½[n+ 1], {[m + 1]),

(Eo, So) = (½[n + m] + 1, ~ l n - m]) . (2.13)

M. Giinavdin et al. / d = 3 AdS and d = 2 con formal superalgebras 435

The other possible lowest states ]n,0)L ® 10, re)R, 10, n)L ® Im,0)R, and 10, n)L ® 10. re)R, all lead to the same irreps as the lowest state In,0)L ® Im,0)R.

To obtain the "massive" UIR's of SU(1,1) we have to take more than two oscillators (a, b) in our construction. If we have 2p oscillator a(r) , b ( r ) ( r = 1 . . . . . p) satisfying

[ b ( r ) , b * ( s ) ] =6rs,

[ a ( r ) , b(s)] = [ a ( r ) , bt(s)] = 0 ,

r , s = l . . . . . p , (2.14)

then the SU(1, 1) generators are given as:

L' = a ¢.b t-~ Y'~at(r)bt(r), r

L = a . b ~ Y ' ~ a ( r ) b ( r ) , r

L ° = ~ ( a ~ . a + b . b * ) = - ~ E ( a * ( r ) a ( r ) + b ( r ) b * ( r ) ) . (2.15) r

The possible lowest states for the construction of the unitary representations of SU(1.1) are much richer for p > 1. The states in the Fock space that are annihilated by L are in general a linear combination of the states of the form:

[a*(1)]"'[a*(2)] "2 . . . [a*(p)]"P[O), (2.16a)

[b*(1 )] "' [ b'(2)] "2... [b*( p )] "'] O), (2.16b)

[ ~'/'~'( rl ) ] . . . . [ (.~[ ( r2 ) ] . . . . . . . [bt(sl) ] ..... . . . [bt(sp)] '*P[O), (2.16c)

where none of the 6 are equal to any of the s i. To obtain a unitary irreducible representation the lowest state must have a definite U(1) charge (10).

if we have an odd number (2p + 1) of oscillators a(r) , b ( r ) and c then the SU(I. 1 ) generators are simply:

L + = a*. b* + ~_c*c*,

L - = a . b + ~_cc,

L ° = { [ a * . a + b . b t + ~(ctc + cc*)] . (2.17)

436 M. Gff, avdin et al. / d = t A d S aJu/ ~/ = 2 cmfformal .superalgehras

The possible lowest states can then be obtained by tensoring the states in (2.16) with the c-boson vacuum 10} and the one-boson state c*lO ).

Again to obtain the massive irreps of SO(2, 2) we need to tensor the massive irreps of SU(1,1)L and S U(1,1)w It is a trivial exercise to determine the labels (E o, So) of the corresponding UIR's. Here we shall give one example to illustrate the difference with the massless case. If we denote an SU(1,1) lowest state of the type [a+(l)]"10)

as In, 0) we have for a lowest state of the type I n, 0)L @ I m, 0)R of SO(2, 2) the following (Eo, So) labels:

( Eo, So) = ( {_[n + m] + p, {_[n- m]) for 2p oscillators,

(Eo, S o ) = ( ½ [ n + m ] + p + ½ , ~ [ n - m ] ) for (2p + 1)oscillators, (2.18)

comparing (2.18) with (2.13) we see that they have the same spin S o but the AdS energies E o differ from the rnassless case by (p - 1)(p > 1) or ( p - ½)(p > 1) units depending on whether we have an even or odd number of oscillators, respectively.

3. Positive energy unitary supermultiplets of AdS 3 or Con 2 superalgebras

The method we shall use for constructing the positive energy UIR's of d = 3 AdS or d = 2 conformal superalgehras is the oscillator method that has been developed in refs. [4, 5]. The Lie superalgebras L that have positive energy unitary representa- tions have a Jordan decomposition (three-grading) with respect to their maximal compact subsuperalgebras

L = L - I ~ L ° ~ L +1 (3.1)

The first step in the construction of positive energy UIR's is to realize the Lie superalgebra L as bilinears of bosonic and fermionic oscillators. Then in the super Fock space of these oscillators (obtained by tensoring Fock spaces of individual oscillators) one looks for states ]~2~ that are annihilated by all the operators in the L - l space and which transform irreducibly under the maximal compact subsuperal- gebra L °. Then starting from the state ]~) which we call the ground state and acting on it repeatedly by the operators in the L *~ space we generate an infinite set of states that form the basis of a positive energy UIR of L. The irreducibility of the representation of L follows from the irreducibility of [~2) under L °. The positive energy UIR's are all of the lowest weight type [4, 5].

Contrary to the case of ordinary non-compact Lie groups, a non-compact supergroup may have a Jordan decomposition with respect to more than one of its maximal compact subsupergroups. For example, the non-compact superalgebra SU(n, re~p) has a Jordan decomposition with respect to its maximal compact subsuperaigebras SU(n/k) x SU(m/p - k) × U(1) where k = 1 . . . . . p. On the other

M. Giinavdm et al. / d = 3 A d S and d = 2 col!formal superalgehras 437

hand the superalgebra Osp(2n/2m,R) has a Jordan decomposition only with respect to its subsuperalgebra SU(n/m) × U(1).

With the exception of G(3) all the Lie superalgebras listed in (1.3) have a Jordan decomposition with respect to a certain maximal compact subsuperalgebra. Thus the oscillator method is not applicable to the construction of positive energy UIR's of G(3). This might simply be a reflection of the non-existence of positive energy U 1R's of G(3). Below we list the maximal compact subsuperalgebras with respect to which the Lie superalgebras in (1.3) have a Jordan decomposition:

(i) Osp(2N/2,R) D SU(N/1 ) x U(1),

(ii) SU(N/1 ,1) D SU(N/1 ) × U(1),

(iii) Osp(4*/2N) D S U ( 2 / N ) x U(1),

{v) F(4) D Osp(2*/4) × U(1),

(vi) D(2,1; a) D SU(2/1) × U(1) (3.2)

We should note that the Lie superalgebras Osp(2N + 1 /2 ,R) do not have a Jordan structure with respect to a maximal compact subsuperalgebra. However they have a Jordan structure with respect to the non-compact subalgebra Osp(2N - 1/2, R) x U(1).

4. The unitary supermultiplets of S U ( 1 , 1 / N ) x SU(I, 1 / M )

The Lie superalgebra SU(1,1/N) has a Jordan decomposition with respect to its subsupergroup SU( I /N) x U(1). It can be realized as bilinears of oscillators &4 and 7/ transforming in the fundamental representation of SU(1/N) and U(1), respec- tively.

(al ~A = a i ' 'r/= b ,

where

(or) ai = i t ' ~/t = b t , (4.1)

[a, a*] = [b, b*] = 1,

{ai 'aJ} =8('/' i , j = I , 2 , . . . N ,

{ el, a , } = {c~', e i } = O. (4.2)

438 M. Giinal,dm et al. / d = 3 AdS and d = 2 conformal superalgebras

Then, as bilinears of ~ and "q the superalgebra $U(1, l / N ) has the realization

(4.3)

Note that the superalgebra S U ( 1 , 1 / N ) can not be realized as bilinears of a single set of superoscillators ~ alone. Thus SU(1, l / N ) has no singleton irreps.

Any state of the form:

,~A~'B... ~cl0 ) (4.4a)

and of the form

(TI*)"I0) (4.4b)

is annihilated by the operators ~A~I in the L -1 space. By starting from any linear combination 1~2) of these states that transforms irreducibly under L ° = S U ( 1 / N ) X U(1) and acting on I~) repeatedly with operators ~A~t of the L ÷ space we generate the basis of a "massless" UIR of SU(1, l / N ) . The bosonic operator atb t of the L + space gives the space-time excitations by acting on a lowest state of an irrep of SU(1,1). The fermionic operator Q' = aib t acts as the supersymmetry generator and interpolates between states belonging to different irreps of SU(1, 1).

For example starting from the ground state 10) we find that the action of Qi generates new states which are lowest states of different UIR's of SU(1, 1) and they have different SU(N) transformation properties. In table 1 we summarize the full

TABLE 1 The full U( N J = SU(N) × U(1 ]~. and SU(1,1) content of the "massless" UIR of SU(1,1/N) obtained

from the ground state 10): the internal U(1),. quantum number y is simply given by the number of boxes in the SU(N) Young tableaux

Lowest State U(N ) -~0 -Y

I 0> 1 1]2 0

O~10> [ ] 1 1

QiQj [0> ~ 3/2 2

Q I1Q 12 • • • Q% 10> N 1/2(N + 1) N

M. Giinardm et al. / d = 3 A d S and d = 2 t in , formal superalgehras

TABI.E 2 The massless UIR of SU(1,1/N) oblained from the ground slate (h i" )"'10>

Lowest State SU(N) -~o Y

(bt) m 10> 1 1/2(m + 1) 0

Oi(bt)m 10> [ ] 1/2(m + 2) 1

o iQ i (b t )m 10> ~ 1/2(m + 3) 2

439

Q'~ • • . O IN (bt) m (0> N 1/2(m + N + 1) N

TABLE 3 The "massless" UIR of SU(1,1/N ) obtained from the ground state ~",~4-~...U,,,Io> =(a~)"'10> e t a ~) .... ~ i l0> • . . . ~a',...~',,,10>

Lowest State SU (N) -~0

(at) m t0> 1 1/2(m + 1)

(at)m -1 o~i /0> [ ] m/2

~ ...~ 10> m 1/2

. . i O ~ l ...o~ = IO> m+1 I

QJl ...QJN moil ...Qim 10> N 1/2(N - m + 1)

m

(m+ 11

N

440 M. Giinavdin et al. / d = 3 A d S and d = 2 conforma/ superalgebras

TABLE 4 The massless unitary, supermultiplet of SU(1, l / N ) L ® SU(1, l / M ) n for the ground state t0)L ® 10) n

Lowest State SU(N) X SU(M) _E 0 _S O

10> L ® Jo> R (1,1) 1 0

Q[ f0> L ® 10> R ([],1) 3/2 1/2

j0> L ® Q~I0> R (1,t~) 3/2 -1 /2

o~o~10>, ® I o > . I~,11 2 1

Io>, ® o~ oJ. io> R tl,E~) 2 -1

i i N Q~...Q O> L® I0~ R

J1 iM 10> L ® QR...QR I0>R

( ~ } N,1)

(1 ,~} M)

1/2(N + 2) N/2

1/2(M + 2) - M / 2

QL...OLf0>, ®Q~...Q~I0>R ( N, M) 1/2iN+M+2) 1/2(N--U)

TABLE 5 The unitary supermultiplets of SU(1 ,1 /N ) with the ground state 10) for p pairs of oscillators: for p = 1

we have a "massless" supermultiplet and for p > 1 we get massive supermultiplets

Lowest State SU (N) "120 Y

10> 1 1/2 p 0

Qil0~> [ ] 1/2 (p + 1) 1

QiQJl0 > ~ 1/2 (p + 2) 2

Q I1Q 12 . . . Q tNI0> N 1/2(p+N) N

M. Giinavdm et al. / d = 3 AdS and d = 2 conformal superalgebras 441

content of the UIR obtained from the ground state 10) by listing all the SU(1,1) lowest states and their U(N) transformation properties.

In tables 2 and 3 we list the U(N) and SU(1,1) content of the "massless" UIR's of SU(1,1 /N) obtained from the ground states (b*)"'10) and ~A,... (A,,I0), respec- tively.

The massless UIR's of an AdS 3 supergroup of the form SU(1 ,1 /N)L® SU(1 ,1 /M) R are obtained by tensoring the massless UIR's of each factor. In table 4 we give the massless unitary supermultiplet of SU(1,1/N)L ® SU(1, 1 /M)R corre- sponding to the ground s t a t e I 0 ) L ® 10)R.

To obtain the massive unitary supermultiplets of the AdS 3 supergroup SU(1 ,1 /N) L ® SU(1,1/M) Rwe need more than one pair of oscillators ~A and rt in realizing the Lie superalgebras SU(1,1 /N) as bilinears:

P L - = ~ A - ~ E ~ a ( r ) ~ ( r ) ,

P

L + = ~a.~, = ~ ~a(r)71,(r), r ~ l

L 0 = ~A. ~B ~ r/. •*. (4.5)

The "mass" of the AdS 3 fields belonging to the massive supermultiplets depends on the number of pairs p. In table 5 we give the unitary supermultiplet of SU(1, l /N) for the ground state ]0) with p pairs of oscillators.

5. The unitary supermultiplets of Osp(2 N~ 2, I])

The non-compact supergroup Osp(2N/2,R) has a Jordan structure with respect to its maximal compact subsupergroup SU(1/N) × U(1). To construct the singleton irrep of Osp(2N/2,R) we realize its superalgebra L as bilinears of a single super-oscillator/~A transforming in the fundamental representation of U(1/N) :

.=(o,) ~A ~--" Of i , Oti

[a, a*] = 1, { a i , a ' } -- 8/. (5.1)

There exist only two states in the super Fock space of ~5 A which are annihilated by the operators ~,~B belonging to the L -x space, namely the vacuum [0) and the state ~A 10). Since they transform irreducibly under L ° = U ( 1 / N ) they can be used

442 M. (;iinavdi, et aL / d = 3 A d S aml d = 2 coqf imnul szqwralgehras

as the ground state for the construction of a UIR of Osp(2N/2,•). The fermionic piece Q'=ato/ of the L +] space acts as the supersymmetry generator and the bosonic piece contains the "raising" operators a*a ~ and aiaj of SU(1,1) and SO(2n), respectively. Thus starting from the vacuum as the ground state we generate the following singleton supermultiplet:

Lowest states SO(2n) l o

1 Io) (oo... o]o) O'lo) (oo...oo]) ~ (5.2)

where we use the Dynkin labels for the irreps of SO(2N). If we start from the ground state ,~'410)= a*l 0) • cd[0 ) we find the singleton

supermultiplet:

Lowest states SO(2n) l o

3 a*lO> (00...010)

,~'10) (oo... ool) ~ (5.3)

Note that the two singleton supermultiplets (5.2) and (5.3) differ by a permuta- tion of the spinor representations (0... 010) and (00... 01) of SO(2N). To obtain the singleton supermultiplets of an AdS 3 supergroup Osp(2 N/2, R) L × Osp(2 M/2, R) r one has to tensor the singletons of each factor. In tables 6a and b we give two of the singleton irreps of Osp(2N/2,R)LXOSp(2M/2,R)R. The other two singleton irreps can be obtained from tables 6a and b by a simple permutation of the factors and the appropriate changes in the S O quantum numbers.

To construct the "massless" UIR's of Osp(2N/2,R) we need a pair of super oscillators ,~A and rF 4 transforming in the fundamental representation of U(1/N).

TABLE 6a Singleton supermultiplet of Osp(2 N/2 , R ) c × Osp(2 M / 2 , R ) r

with the ground state [0)c × 10)R

Lowest states SO(2 N ) × SO(2 M ) L~) S~)

Io>~, x Io)r (o... lo)(o.., lo) ~ o Q[.I O) × [O)r (0 . . .01)(0 . . . 10) 1

[0)t. x Q h l 0 ) r (0 . . .10)(0. . .01) 1 -

Q~.IO)L × O~JO)r (o.. .Ol)(O...O1) 3 2 0

M. Giinm,dm et al. / d = 3 AdS and d = 2 conformal superalgebras 443

TABLE 6b Singleton supermultiplet of Osp(2 N/2, R) L × Osp(2 M/2 , R ) R with the ground state [ 0) L ® ~ ~ I 0) R

Lowest states SO(2N) X SO(2 M) E 0 So

IO) L × a~10) ~ (0... 10)(0... 10) 1 - ~2

IO)L X ~I0)R (0...10)(0...01) { 0 Q~IO)L X a~lO)p. " (0...01)(0...10) _3 0 QL Io)t x ,& Io)R (o... oh(o...o1~ 1 '~

The generators of Osp(2N/2,R) are now realized as bilinears of ~ and rt:

L = + • + nBn") • ( ¢ " , : + n " ¢ " ) ,

L + ' = + ,7 , ' ;B ) . (5 .4)

Any state of the form

or of the form

(A~s . . . ( c l 0 ) (5.5a)

~IA~I~... ~lcl0) (5.5b)

is annihilated by the operators belonging to the L -~ space. By choosing as the ground state a linear combination of these states that transform irreducibly under U ( 1 / N ) subgroup one can generate a massless UIR of Osp(2N/2, R). To obtain the "massive" unitary representations of Osp(2N/2, R) we have to realize its superalge- bra as bilinears of more than two super-oscillators transforming in the fundamental representation of the maximal compact subsupergroup U ( I / N ) . The massless and massive UIR's of the d = 3 AdS supergroups O s p ( 2 N / 2 , R ) × Osp(2M/2,R) are then simply obtained by tensoring the corresponding UIRs of each factor.

Let us now apply this general construction to the superalgebra Osp(8/2,R). Its maximal compact subsupergroup is U(1/4). Consider now a pair of oscillators ~A and ~/A transforming in the fundamental representation of U(1/4).

where the ct i and fli are the fermionic oscillators. The bosonic part of the L +l operator (~A~/s+ r/A~s) contains the SU(1,1) raising operator atb t and the SO(8) raising operator aPfl Jl. The fermionic piece is simply the supersymmetry operator

444 M. Gffnqvdin et al. / d = 3 AdS" and d = 2 conjbrma/ superalgehras

TABLe 7 The massless unitary supermuhiplet of Osp(8 /2 ,~ ) with the ground state 10)

Lowest states SO(8) l o

Q'I0) 56~ 1 Q'Qi]O ) 28 3

Q'Q'Qt]O) 8~ 2

Q'Q/QaQIIO ) 1

TABLE 8 The unitary supermultiplet of O,Sp(8/2, R)L X Sp(2.R)r with the ground state 10)e × [O) r

Lowest states SO(8) E o S'~

]0)t. X I0)R 35,. 1 0

QI.10)L × 10)r 56~ ~

QI.Q~[O)L × f0>a 28 2 1

Q] Q[ Q~.Q~I[O)L × 10)R 1 3 2

Q i = a~//i+ b*aq By acting on a ground state 112) with Qi we generate new states which are the lowest states of both the SU(1, 1) and the SO(8). The full content of the UIR determined by the ground state can then be read off from these lowest states. In table 7 we give the UIR with the ground state ]0)*. In table 7 and the later tables for simplicity we shall denote the lowest states by the supersymmetry generator Qi acting on the ground state [I2), e.g. QqI2), QiQj[~) etc. We should warn the reader that Q~... QklI2) may contain, in addition to a lowest state of the even subgroup, states which are "excitations" of some other lowest states of the even subgroup. We shall implicitly assume that we are only considering the true lowest states and not the excitations. For details on this point see refs. [5].

Consider now the AdS 3 superalgebra O s p ( 8 / 2 , R ) L × S p ( 2 , R) R, whose UIR determined by the ground state 10)c x 10)r is given in table 8. The corresponding supermultiplet of AdS fields is a short one in the sense of ref. [10]. In fact all short AdS supermultiplets (massive as well as massless) can be obtained by choosing as the ground state the Fock vacuum [5]. The AdS 3 supergroup Osp(8/2 , R ) e X Sp(2, R)~ has an important long massless supermultiplet whose SO(8) and spin content is identical to the decomposition of the N = 8 Poincar4 supermultiplet in

* Here we follow the conventions of the second reference in [5] for the labelling of SO(8) representa- tions.

M. (;finavdin et al. / d = 3 AdS a m i d = 2 coqfi>rmal superalgehras 445

d = 4 with respect to SO(8) and helicity. This unitary supermultiplet is obtained

f r o m the g r o u n d state

.~ 4

In table 9 we give the SO(8) and SO(2, 2) content of this supermultiplet. The massive unitary supermultiplets of Osp(8/2,R) can be obtained by taking

m o r e than one pa i r of super -osc i l l a to r s t r a n s f o r m i n g in the f u n d a m e n t a l r e p r e s e n t a -

t ion o f S U ( 1 / N ) . In table 10 we give the u n i t a r y s u p e r m u l t i p l e t s of O s p ( 8 / 2 , R )

d e t e r m i n e d by the g round s ta te 10).

U s i n g t ab le 10 we can o b t a i n the shor t u n i t a r y s u p e r m u l t i p l e t s of d = 3 an t i -

de S i t t e r s u p e r g r o u p O s p ( 8 / 2 , R ) L ® S p ( 2 , R ) R o b t a i n e d by c h o o s i n g the g r o u n d

T A B L E 9

The long massless unitary supermultiplet of Osp(8/2, N)t. × Sp(2, R) R ,4 B ( D + with the ground state ~t.~t.~t.~t, 10>t. ® (at)410)R

Lowest states SO(8) E o ,%

,~L,~?~f-I. Io> L x 14 )'1o> R ~L,~?4.41o> ~ × (4)41o>

.,i a( ( . l )-'I0>L × ("~)"I0>R i 3

a t ( a l . ) 0 ) 1 × ( a ~ ) 4 1 0 ) R

I-l ?lOb. x (.~ ?10>~ ,L )~.{!fl/' l o> t. × ( 4 )~1o> R

(-l)'lq{o>,. ×(4)~{o>R ~" 6 * I (aL) flt.flk[O) X (a~)"iO)R

"~" N i / A / i" 4 (a,.) fl,.flLfl,.fl,.lO>E x(a,~) }O>R

1 3 - 2

28 4 - 1

35~ 5 0 35, 5 0

',' '2 28 6 1

1 7 2

TABLE 10 The unitary supermultiplets of Osp ( 8 / 2 , R ) for the ground state I 0> with n superoscillators. F o r . = 2 we get a massless supermultiplet and for n = 1 one of the singleton supermultiplets

(the last 3 rows arc absent in this case). For n > 2 we get the short unitary supermultiplets.

Lowest states so(g) / o

IO) (n.O.O.O) 14n

Q'IO) <, - 1.0. t.O) '~(, + 27

O'O'10> (n - 2 , 1 , 0 , 0 ) ~ (n + 4)

Q'Q'Q~ JO> (,1 - 2,0,0,1) ~4(n + 6)

Q'Q'Q~QIJO) (,1 - 2,0,0,0) ~4(n + 8)

4 4 6 M. Giinto,din et aL / d = 3 A d S and d = 2 conformal superalgebras

TABLE 11 The short unitary supermultiplets of Osp(8/2, ~ ) L ® Sp(2, R) r obtained

from the ground state [0)L @ 10)R

Lowest states SO(8) Eo So

10>L X IO) R ( 1 1 , 0 , 0 , 0 ) 1 ~n 0 Q[ IO)L X IO)r (,1 - 1,0,1.0) ]2( n + 2) ,

OLQ(~ Io)t_ × Io)R (,1- 2, ],o,o) 1~(,1 + 4) 1 Q,,q/r~ 0)LX[0>r (n-2,0,O, 1) ~(n+6) 1. t~" i. ~" - - 2

, , ~ / ~ ( n + 8 ) 2 Qt.QL QLQI. IO)L × IO)r ( n - 2 , 0 , 0 , 0 )

The number n denotes the number of super-oscillators. For n = 1 only the first two rows exist and we get a singleton supermultiplet. For n = 2 we get the short massless supermultiplet. For n > 2 we obtain the short massive supermultiplets.

state I0)L@ I0)R. They are listed in table 11. We should note that the short supermultiplets given in table 11 give us the full spectrum of the compactification of the ten-dimensional simple supergravity on the seven sphere to three dimensions with the corresponding vacuum having Osp(8/2, R)L ® Sp(2, R)R symmetry. Inter- estingly enough, the chirality of the d = 10 supergravity is reflected in the d = 3 theory by the fact that the spectrum consists only of positive spin (helicity) states. Again we expect the singleton supermultiplets to decouple from the spectrum being pure gauge modes [5].

By choosing all possible ground states annihilated by L-1 operators and trans- forming irreducibly under L ° (with an arbitrary number of super-oscillators) one can similarly construct all the positive energy UIR's of Osp(8/2 ,R) , Osp(8 /2 ,R)L ® Sp(2, R)R and Osp(8/2, R)L® Osp(8/2, R) R. The extension of the method to construct all the positive energy UIR's of the superalgebras Osp(2N/2, R)L® Osp(2 M / 2 , R)R is straightforward. Using the results of the previous section one can easily construct the UIR's of superalgebras of the form S U ( 1 , 1 / N ) L ® O s p ( 2 M / 2 , R ) R o r of the form Osp(2N/2,ff~)L ® SU(1, 1 /M)R.

6. Discussion

In the above we have classified the d = 3 anti-de Sitter (or d = 2 conformal) superalgebras and have given a general method for the construction of positive energy unitary supermultiplets of these superalgebras. We studied in detail the unitary supermultiplets of the superalgebras S U ( 1 , 1 / N ) L ® S U ( 1 , 1 / M ) R and Osp(8 /2 , • ) L ® SU(1,1)R. Since all the positive energy (lowest weight) UIR's of non-compact groups are obtainable by the oscillator method and since the unitary finite dimensional representations of compact groups are all of the lowest weight

M. GtJ'nuvdm et al. / d = 3 A d S and d = 2 co~l[ornlal superalgebras 447

type we expect to be able to ob t a in all the posi t ive energy U I R ' s of n o n - c o m p a c t

supe rg roups by the oscil lator method . The extension of our cons t ruc t ion to the case

of O s p ( 4 * / 2 N ) is s t ra ightforward. However the excep t iona l cases F(4), G(3) and

D~(2, 1; c~) require special t rea tment . The un i ta ry supermul t ip le t s of the excep t iona l

supe ra lgebras as well as those of O s p ( 2 N + 1 / 2 , R) will be the subjec t of a sepa ra te

s tudy.

Two of us (M.G. and G.S.) wou ld like to thank the organizers of the C a m b r i d g e

W o r k s h o p (1985) on Supersymmet ry and its app l i ca t ions for p rov id ing the s t imula t -

ing a tmosphe re in which this work was carr ied out.

Note added

Af te r the comple t ion of this work we became aware of two very recent works

[12, 13] on c o n f o r m a l supergravi ty in d = 2.

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I. Bars and M. Gi~naydin, Comm. Math. Phys. 91 (1983) 21 [5] M. Ghnaydin, P. van Nieuwenhuizen and N.P. Warner, Nucl. Phys. B255 (1985) 63:

M. Gi~naydin and N.P. Warner, Nucl. Phys. B272 (1986) 99, M. Gi~naydin and N. Marcus, Class. Quant. Gravity, 2 (1985) Lll

[6] S.K. Han, 1.G. Koh and H.W. Lee, Sogang preprint 113 (1985) [7] C. Fronsdal, Phys. Rev. D26 (1982) 1988;

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[10] D. Freedman and H. Nicolai, Nucl. Phys. B237 (1984) 342 [11] M. Ghnaydin, B.E.W. Nilsson, G. Sierra and P.K. Townsend, CERN preprint TH 4392/86 [12] P. van Nieuwenhuizen, Stony Brook preprint ITP-SF-85-72 (1985) [13] A. Salam and E. Sezgin, Trieste preprint (1985)